Electrical network



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o. J. 20am. M1295??? ELECTRICAL NETWORKS Filed Sept. 9, '1926 2 Sheets-Sheet 2 moRNEx/s.

@TTU (il. ZOBETJ, 0F N YORK, N. Y., SSIGOR T0 AMERIGAN TELEPHONE AND TELE- GRAPH COMPNY, A CORPORATIQN OF NEW YORK ELECTRICAL NETWORK.

.application tiled ySeptember 9, 1926. Serial No. it.

The principal object of the present invention is to rovide an electrical network of definite and) practicable design that shall simy ulate the impedance of a given smooth transmission line or any electrical conductor system having an impedance-frequency characteristic similar to such a line.

The invention will be better understood by reference to the accompanying drawing in which Figures 1 to 3 and Figs. 2, 2l and 3 are circuit drawings to assist in explaining the invention; Figs. 4 and 10 represent two forms of networks embodying the ideas of my invention, Figs. 5 to 9 are equivalents of Fig. fi; Figs. 11 to 15 are equivalents of Fig. 10;

Figs. 16 and 17 are networks on which actual calculations have been made for illustrative purposes and Figs. 18 and 19 are curves showing the degree to which the calculated networks of Figs. 16 and 17 agree with the smooth line they were intended to simulate.

Tn the consideration of wave filters and artificialJ lines it has become useful to speak oi electrical structures known as ladder-type structures and shown in a generalized form' in Fig. 1 of the drawing as consisting of an iterative impedance, each section comprising series impedance and shunt impedance. The porperties of these structures are described broadly in various places such as Transmission Circuits for Telephonie Communication, by K. S. Johnson, p. 121 et seq., where they are classified as mid-series and mid-shunt iterative impedances. These two forms are shown generically in Figs. 2 and 3 from which it is seen that Fig. 2 is made up of a succession of T structures and Fig. 3 of a succession ot 1r structures. The extent to which such structures may be made to simulate a smooth line depends on the number of sections used but it is known that certainof these two distinct ladder-type structures have at some termination in each a characteristic impedance equal to that oi? a smooth line, such as an open wire or a cable signaling line, namely .uw and propagation constants, which can be vared over a wide range, and differing, in general, from the propagation constant of the smooth line. In one this termination is at mld-series, inthe other, it is at mid-shunt, so that mid-series and mid-shunt sections, respectively, have the characteristic impedance T. -`Where R/LG/C, as in most telephone lines, the mid-series section has a series iinpedance of resistance and a shunt impedance of two distinct resistanccs and one capacity, as shown in-Fig. 2. In the mid-shunt section the positions of these types of impedances are interchanged, as shown in Fig. 3a. Where llt/L15( ir/C, the sections have the same forms but with capacities replaced by inductances, as shown in Fig. 2b which corresponds to Fig. 2a.

As pointed out above, the degree of simulation obtainable in any case, depends on the number of sections, a, in such ladder structures and becomes exact for all frequencies only when a is iniinity. However, a certain degree of simu lation is obtained even with one or with two sections and in both of these cases a substitute and equivalent network can be found using a smaller number of resistanceelements. The transformations for finding such substitute networks for a few simple Lcases are given in my article on Theory and design of uniform and composite waveilters, Bell System Technical Journal, January 1923, p. 45,. As an illustration, let us consider first the important case where R/LG/C. Take one section of either kind above and terminate one end by a resistance of any magnitude thus giving a two-terminal impedance network. By certain impedance transformations, this network, as far as its driving-point impedance is concerned, can be reduced to a minimum of three elements, two resistances and one capacity, having the form either of a resistance in parallel with one branch of series resistance and Capacity, or a resistance in series with one branch of parallel resistance and capacity.

Next take two sections of either kind, or both, join themtogether, and then terminate by a resistance. This two-terminal network can be reduced, without altering its impedance, to one of live elements, three resistances and two capacities. One form it may have is a resistance in parallel with two branches or i from series resistance and capacity, as shown in Fi g. el. Another form is that of a resistance in series with two branches of parallel resistance and capacity, as shown in Fig. l0. 5 to 9 are live other impedance forms equivalent to and igs. ll to i5 are tive equivalent to l0.

lilith tnree sections, the reduced network would have seven elements. rlie forms corresponding to the two above have three branches in each case. llie may continue this 'to any number sections and arrive at the following conclusion:

The driving-point impedance of n sections having the characteristic impedance l (midseries, mid-shunt, or both,) and terminated byany resistance is identically that of a reduced network having 2nd-l elements, (ni-l) resistances and n capacities. @ne form which these elements may have is a resistance in parallel with n branches of series resistance and capacity. Another form is that of a resistance in series with a branches of parallel resistance and capacity. For a greater than l there are other equivalent forms, the number increasing` very rapidly with n.

. For the case wnere lt/LLG/C the same general conclusions apply wherein all capacities are replaced by inductances.

ln all these reduced networ rs the number of resistances has been diminished to a minimum, a fact of practical importance.

Since each sect-ion has attenuation, the larger the number of sections taken the less the effect due to the imperfect resistance termination and the more accurately does the driving-point impedance approach that of the desired characteristic impedance K. Hence, obviously, the more parameters in the reduced networks the better the possible iinpedance approximation.

The above was based upon the assumption that the line parameters R, L, G, and C are independent of frequency, as that is the basis of the ladder-type structures. However, the forms of the reduced networks would still be valid even when the line parameters varied somewhat with frequency, as they do. Such variations are no hindrance to the design p method given below.

Met/lod of design..

rThe impedance of networks will, in general, depend on the frequency and for any given network containing reactances may be will@ l-l-ijfal inl-ifm,

This gives an impedance expression of the .E iorm which must correspond also to any of the other equivalent structures. Here r and c@ are the components of the impedance to be simulated. lin both Equations (2) and the zs and Zis and the nfs and Bs are constants for a given network (of which Fig. t would be an example with W22) also, in these equations f is the frequency, an inde pendent variable, and i= ,/-l The impedance e, according to Equation (3), is the ratio of two polynomials of nth degree in (if), with 272+ l coemcients which occur linearly.

` rlhe important observation tomake here is that these coefficients, A0, Bn, which are functions of the network elements, are eXactly equal in number to the number, Qn-i-l, of those elements. For example, in Fig. 1i, the As and the Bs are functions of the Rs and the Cs and there are five As and Bs in Equation when 71:2 and there are ve Rs and Cs in Fig. 4. Hence, in general, if these coefficients, the As and the Bs, can be dotermined, the network elements (for example, the Rs and Cs of Fig. t) are fixed in terms of them.

The method of deriving these coefficients, and thus the network elements, directly from the networks impedances at arbitrary frequencies is as follows: First, clear E nation (3) of fractions and group separate y only7 real and imaginary parts containing the coefficients in the left-hand member which must equal the right, r-i-r'. Assuming merely for convenience that n is even, the form of this equation will be [.a0-f2a2+ if)fa+fael+f2ffn2+ (rfv-ra] +iva, treffend-fre,+fan2+ (afremmen. (i)

, sults, however.

r iF rom these coefficients the network elements maar?? Since the real and imaginary parts must sep arately satisfy the equation we have the equations (simultaneous at any frequency) Ao-,fzAz-t (f)rB==r, v

' ffll-i- (f)"w=m. (5)

At any given frequency these are linear equations in the coefficients A0, Bn. Remembering that r and are the resistance and reactance components of the imped ance of the smooth line or other conductor system to be simulated, they will have definite values which may be ascertained (or computed) for any given value or values ot the it'requency f. Accordingly, tix r and w, as obtained by measurements er calculations, at a number of frequencies so as to obtain 2nd-l consistent equations. We then have a system ot simultaneous linear equations from which to obtain the coetlicients by ordinary methods.

UJICG.

sion given hy equations can be found; they will be physical provided the network is capable of having the impedances assumed. I

@ther ways ot setting up the system of equations would be to consider (5) as non-simultaneous equations or use their sum or diterence. While such assumptions would lead to solutions tor the coefficients there would he no guarantee that the resulting network would have any ot the impedance components assumed. rlhey might give goed requency, in addition to the impedances at an two intermediate frequencies f1 and f2.

Lower frequency range..

ylhe above inode of procedure tor fitting a network to an impedance characteristic which it can have 1s new, but it is straightforward and one which is obviously valid whenever The data are yto=0, ro-t-O.,

the number of independent coefficients ot llutii'n powers oit in the impedance expression 2, ff'zfi@ ma r equals the number oit independent networir elements. This condition is known to hold Substitutign in gives for a large variety ot networks5 but not tor some others i T0- gni ses

The method just described can now he used to obtain design formulae and, as an illustra- The solution ot (9) may be put as "the coecients g, t, m, a, and e are all qpositive in a physical structure, ff

tiem-the will be used tor the 5-element struetures of igs. Il to l5. These networks will be considered as sufficiently typical and are also practical. They correspond initially to two ladder-type sections terminatedby a resist- '.lhe networks have an impedance expres- Clearing fractions gives the simultaneous 'lwo sets ot formulae will be derived for these eoecents; one containing the impedance at zero frequency2 the other that at iniinite fre- Urgcer tregua/w3! mage.

The coeicients are all positive.

lt should be 'pointed out that still tliirtl set ot formulae like (l) and (13) could be obtainecl by taking as the tliircl arbitrary trequency neither zero nor iniinityancl using but one or tlie tWo Equations (7), their sum., or their difference. Other combinations of linear equations may be usecl as the basis for an approximation to the given impedance clata. l@ or exam-ple9 tlie tivo equations of (7) may be used a dili'erent number of times anti evV i different frequencies. Again, We miglit use at the lowest arbitrary frequency the (lili-@erence of Equations (7) at the next liiglier tre quency tlieir sum, and so on alternately until equations have been set up to include data at live arbitrary frequencies. The solutione for all suoli cases are obtained roost reaclily direct fro-ni the linear equations cliosen ami will not be given liere in explicit torno.

Netw wie elemente.

Having determined the coeircients gy Ft, m. 11,7 and s, the network elements are fiund from tlieiu. l-lere it will be sufficient to obtain Such explicit formulae only for ll'gs. fr, l0, l@ and 17 since when tlio elements of these two structures are known tliose of toe otlier equivalent forms can be obtained if clesireri by an application of the equivalent network `transformation formulae of my article in tlie Bell System Journal mentioned` above.

lt one Writes down the impedance of the network of Fig. l or Fig. 16 in terins of its Rs ancl @s and simplifies the resulting eirpression and compares coeficiente Witli Equation (6), the following relations will become apparent: 9': R1 h= QmlPJRzl-REQS m: lau-BRIE R 1059 (le) lor gm (uit) y (le) Wliere 1 l v w) (a :l: a/agm ee)/2. pplcatz'oa of form/alte.

'v-lieg coulcl as Well llave been obtainedA vfrom measurements.

For the frequency rang. 0-5000 cycles/sec. at 20o are very accurately given by R: 10.12-1-.004 L: .00366 henrys,

where R, L,'G and C for the smooth line are constants but is applicable to any line or twoterminal electrical conductor system for which the impedance is a function of frequency and is approximately the same which it would be if those values were constants. In illustration of this statement it may be noticed that in the above Equations (18), R is not a constant but is a function of frequency; however, for frequencies not too great, R has nearly the constant value-10.12; similarly for G. In many cases of a conduc- `tor system the values of R, L, G and C themselves may not be known, as vwhen the system is a composite line or contains intermediate apparatus only the measured impedance may be known, and to proceed with the desi n of the network according to the present dlsclosure it would be necessary to know only the values of 7 and for each of n+1 values of f according to the notation of this specification.

Assuming that it is desired to have a network which shall give particularly good sim-v ulation in the lower frequency range, the data in (8) will be taken as m: .2302. 4. The network elements for Figs. 16 and 17 corres ending to these parameters are from (15) an (17), respectively,

121:1778 ohms, C', :.07860 mf., R,=55,990 ohms, 0 :.5644 mf., (20) 128:1321 ohms,

and.'7

R:747.6 ohms, 0, :47.13 mf., R,=97.38 ohms, 010:1,821 mf., R8:933.0 ohms.

C. the line parameters per loop mile (f/iooo.) +.o32 (f/loco) 2 ohms,

whence g: 1587, n: 9,537.1-10?, 1:75439, 8=1o.1171o, (22) m:.0066904.

These givel I?, :1587 ohms/04 :.4688 mf., R2 :1213 ohms, U5 :.01088 inf., (23) R3 :17260 ohms,

and

Comparison of the impedances, a, of these two different range networks with that, K, of

the smooth line are given in Tables I and II, i

and on Figs. 18. and 19. A column is added in the tables glving the percentage impedance-departure, P, as computed from z K P E (25) Table I .-Lowerlrcqucucf/ frange.

Percent- Frequcncy Line impedance Network impodare imcyclos/sec. (ohms) ance (ohms) ednnceeparture O 1778 1 0 1773 i 0 0 10 1760 1123.1 1760 1124. 4 .074 20 1713 1232. 7 1713 i232 7 0 40 1575 1380. 7 1578 1385. 1 333 70 1359 12185. 6 1365 ma 2 .41s 100 1104 1405. 5 1104 1405. 0 200 017. 5 1405. "1 918. 2 1374. 7 3. 05 500 727. 0 1'218. 1 77117-11755 8. 86 1000 G80. 0-1115. 6 755. 8 1 90. 0 11. 50 2000 066. 5 1 56.8 740. 7 1 45.3 12. 60 3000 663. 7 1 30.7 V 748. 5 1 30. 2 12. 8O 6000 002.3 1 21.2 747. 0 1 18. l 12. 03 O0 661.3 1 0 747. 0 1 0 13. 05

Ta ble IL Upper frequency ram/e.

Percent- Frequency Line impedance Netwerk impcdngc 1mcycles/sec. (ohms) ance (ohms) ednncccpnrture It will be seen that over the frequency ranges for which the networks are designed their impedance departures are quite small.

The departures of the networks designed for the 'upper fre uency range might be further decreased at t ie-intermediate frequencies by making slight changes in the data or frequencies used.

It is to be understood that the example given is yfor illustrative purposes only, and that if the requirements of simulation are more exacting, then networks of a larger number of elements would be used. However, I have shown how to find the values to be assigned to the (2a-t1) elements which is the, smallest number of elements which can be associated together to be an impedance which is the equivalent at all, frequencies of a ladder-type network of lnl sections with a terminating resistance, each of said sections coinprising four elements as indicated in F igs. 2, 2b or 3a, and which simulates, with a. high order of accuracy, the impedance cliaractei'- istics of a smooth line. Furthermore, I have disclosed a method of design which enables one to design a network of a limited number of elements which shall have the least departures from a smooth line at those frequencies or range of. frequencies where closest matching is desired.

An essential feature of my method resides in using a plurality of eoeiiicients which are related in acomplicated manner to the constants of the network to be designed, but which eoeiicients do not appear as powers or as products of each other. The method of treatment is such as 'to yield simultaneous equations which are linear in these coeiiieients, thus enabling these latter to be readily evaluated. Later, the constants themselves may be obtained from the coeiicients.

As has been shown in this specification and in the drawings, there are numerous equivalent networks that can be designed all in accordance with the principles of my invention to simulate a given smooth line or other twoterminal conductor system. For example, the networks of Figs. 5 to 9 are all precisely equivalent to Fig. 4, and the networks of Figs. 11 to 15 are all precisely equivalent to Fig. 10. For the sake of elearness in this speciication I mention and discuss Fig. .4 principally, with the understanding that it is representative of the whole class of networks that may be designed to simulate a given conductor system such as mentioned above.

What is claimed is:

1. A network of 'a+ 1 parallel branches between a pair of terminals, where a is a positive integer not less than 2, one such branch consisting of a resistance, and each remaining branch consisting of a resistance and a capacity in series, said resistances and capacities having values that make the impedance across said terminals precisely the sameas for a given two-terminal conductor system Iat ln+1 different pre-assigned frequencies,

said conductor system having an impedancefrequency characteristic similar to that of a smooth line, and said network having a close approximation to the saine impedance-frequency characteristic over the frequency range within the eXtreme values of said n+1 dili'erent frequencies.'

2. A network of three parallel branches between a pair of terminals, one such branch consisting of a resistance and each remaining branch consisting of a resistance and a capacity in series, said resistances and capacities having values that make the impedance across said terminals precisely the same as for a given two-terminal conductor system at three different pre-assigned frequencies, said conductor system having an impedancefrcquency characteristic similar to that of a smooth line, and said network having a close approximation to the same impedance-frequency characteristic over the frequency range within the extreme values of said three frequencies.

3. A network of a+ l parallel branches between a pair of terminals, where a is a positive integer not less than 2, one such branch consisting of a resistance, and each remaining branch consisting of a resistance and a capacity in series, said resistances and capacities having values that make tlie impedance across said terminals precisely the same as for a given two-terminal conductor system atA a+ 1 different pre-assigned frequencies, said conductor system having an impedancefrequency characteristic similar to that of a smooth line, and said network having a close approximation to the same impedance-f requency characteristic over the frequency range within the extreme values of said a+ 1 different frequencies, one of said n+1 different frequencies being at an extreme value on the positive number scale.

4. A network of 'n+1 parallel branches between a pair of terminals, where n is a positive integer not less than 2, one suc-h branch consistingr of a resistance, and each remaining branch consisting of a resistance and a capacity in series, said resistances and capacities having values that make the impedance across said terminals precisely the same at all frequencies as for a ladder-type recurrent network of n sections terminated by a resistance and with three resistances and one capacity per section, such ladder-type network when its sections are repeated indefinitely being precisely equivalent to a given two-terminal conductor system at n+1 different pre-assigned frequencies, said conductor system having an impedance-frequency characteristic similar to that of a smooth line.

5. A network of a+ 1 parallel branches between a pair of terminals, where n is a positive integer not less than 2, one such branch consisting of a resistance, and each remaining branch consisting of a resistance and a capacity in series, said resistances and capacities having values typified; bylEquations ('15)v for the case of fn=2.

i6. A network of w+ l resistances and n like reactance elements between a pair of terminals, where n is a positive integer not less than 2, said resistances and reactance elements having values that make the impedance across saicl terminals precisely the same as for a given two-terminal conductor system at n-lf 1 dierent pre-assigned frequencies, said convductor system having an impedance-frequency characteristic' similar to that of a smooth line, and said network having a close approximation to the sanie impedance-frequency characteristicv over the frequency range within the extreme values of said w+ 1 different frequencies.

L A network of three parallel branches between a pair of term'inals, one branch consist ing of a resistance R1, another branch consisting of a resistance R2 and capacity Gg, 1n series, and the remaining branch consisting of a resistance R3 and a capacity U5 in series, the values of said Rte and Cs being in accordance with Equations (15) 8. A network of three parallel branches between a pair of terminals2 one branch consisting of a resistance R1, another branch consisting of a resistance R2 and capacity G4 in series7 and the remaining branch consisting of a resistance R3 and a capacity C5 in series, the values rof saicl Rs and Cs being in accordance with Equations (15) expressed'in terms of g, m, n and s and these being in accordance with Equations (10) expressed in terms of sets of quantities f, 7' and ai, where eachi represents a given frequency and each i" represents a corresponding resistance and each m represents a corresponding reactance, these,` i fs,1^s and s being values for a two-terminal 40 conductor system similar to a smooth line.

1n testimony whereof, ll have signed my name to this specication this 7th day of September, 1926.

,erro af Zonen. 

